It already has. Any tech company that is pre-IPO and still raising funding rounds is a "startup". I'm surprised there hasn't been someone to come up with a separate term for the stage of these kinds of companies.
Still, I think there needs to be a specific term for a company that has recently had a funding round and will likely IPO in the future, like Stripe. That's a different category than a start-up or privately owned company that will never IPO like Koch Inc.